The major factors determining pharmacologic drug responses are the input and disposition rates controlling pharmacokinetics, drug distribution to the site of action (biophase), the mechanism of drug action in altering mediator or receptor levels, and transduction processes which follow the latter. The Principal Investigator has developed a family of four indirect response (IDR) models to account for drug action on the mediators of the response rather than directly causing the response. Because these IDR models along with added complexities currently require analysis using differential equations which often cannot be fully integrated, the Principal Investigator proposes to use advanced methods for calculus and simulation to seek exact or approximate solutions or behaviors for such models in order to yield improved insights and methods for understanding the time course of drug response as related to major mechanisms of action. This project seeks to characterize and quantify the problem of drugs acting by direct and indirect mechanisms. Specific aims include: elucidating properties of IDR when drug is given by short and long-term infusions, identifying parameters determining the linear return rates of responses, using moment analysis to characterize experimental data, dealing with variable baseline behaviors, addition of a precursor compartment to account for tolerance and rebound effects, and applying irreversible rather than reversible inhibition of the response variable. Advanced methods of calculus and simulations will be employed to seek exact or approximate solutions or behaviors for these models, to identify how the onset, extent, return, duration, AUC, and mean times of responses are controlled, to recover parameters more easily from experimental data, and to discriminate among diverse models available to describe various, types of data. These efforts should yield improved insights and methods for understanding the time course of drug responses as related to major mechanisms of action.